Please give me the mathematical proof of the three equations of motion.

Let us suppose that v ₀ is the initial velocity of the particle, at time t = 0, and v be the velocity of the particle at t = t.

The acceleration a of the particle can be written as,

This is the 1^st kinematic equations for uniformly accelerated motion.

Suppose, the particle travels a distance x in the time t.

The area of blue colored part gives the distance covered x.

So,

This is the 2^nd equation.

Now, for the 3^rd equation we shall modify the equation:

Regards